1.

INTRODUCTION

Practical course of science and engineering such as Curriculum Design, Specialty Experiments aims to develop the students’ ability to research, analyse and solve problem, train students’ practical skill and innovation ability. How to assess the students’ performance objectively and exactly in practical course is one of the ubiquitous problems. The traditional way of evaluation mainly adopts final course grade which is determined by assignments and class performance and mark on the final with different percentages. Due to lack of a uniform criterion of justness, traditional evaluation methods is based on teacher’s subjective judgments, and usually results in area indexing of students’ grade is insufficient, and can’t evaluate the students’ performance objectively and exactly. In allusion to this phenomenon, statistics method is used in the comprehensive assessment of practical course in many colleges such as standard deviation, cluster analysis, principal component analysis, and so on^[1-3]. These methods improve the traditional evaluation method, and the effect is good, but the statistics and calculation is complex and needs huge work, which will increase the workload of the teachers, moreover, these methods still can’t avoid subjectivity of teachers. The fuzzy comprehensive evaluation based on the theory of fuzzy mathematics is an effective method which considers a multiplicity of different influence factors and can assess the students’ performance objectively and scientifically ^[4-6].

Here, based on Engineering Education Accreditation Standards and the theory of fuzzy mathematics ^[7-9], on the basis of confirming the evaluation’s factor assembly and concrete evaluation factor, through two layers evaluation, the quantitative comprehensive evaluation model of graduation requirement achievement degree is established.

2.

ESTABLISHMENT OF AHP-Fuzzy EVALUATION MODEL

2.2

Determination of weight vector W

The traditional methods to determine the weight of each attribute include the expert estimates method, the weight statistic method, the frequency statistic method and so on ^[10]. Here, fuzzy analytic hierarchy process is applied to determine the weight of the first achievement degree evaluation’s factor and the second achievement degree evaluation’s factor. As the following step:

Step 1: The nine-point scale fuzzy analytical hierarchy process is used to construct the fuzzy judgment matrix: X = (x_ij)_{n x n} (i, j=1,2,···,n). The meaning of each scale measurement is explained in Table 2.

Table 2.

Scale of preference between two elements

Scale	Definition	Explanation
1	Equally preferred	Two activities contribute equally to the objective
3	Moderately	Experience and judgement slightly favour one activity over another
5	Strongly	Experience and judgement strongly or essentially favour one activity over another
7	Very strongly	An activity is strongly favoured over another and its dominance demonstrated in practice
9	Extremely	The evidence favouring one activity over another is of the highest degree possible of affirmation
2, 4, 6, 8	Intermediate values	Used to represent compromise between the preferences listed above
1, 1/2, 1/3, ···, 1/9	Reciprocals	Reciprocals for inverse comparison

Step 2: The pair-wise comparison matrix, X = (x_ij)_{n x n} (i, j=1, 2, ···, n), the eigenvector is given by equation (1):

for all i = 1, 2,…, n

Step 3:The eigenvector needs to be normalized, and the weights are computed by equation (2):

for all j= 1, 2,…, n

Step 4: Measurement of consistency: the consistency ratio is calculated as per the following steps:

• (1) Calculate the eigenvector or the relative weights and λ_max for each matrix of order .

• (2) Compute the consistency index for each matrix of order n by the formulae: .

• (3) The consistency ratio is then calculated using the formulae: .

where RI is a known random consistency index obtained from a large number of simulation runs and varies depending upon the order of matrix. Table 3 shows the value of the random consistency index (RI) for matrices of order 1 to 10.

Table 3.

Average random index (RI) based on matrix size

N	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.52	0.89	1.12	1.36	1.41	1.46	1.49	1.52

If the value of CR is less than 0.1, it implies that the evaluation within the matrix is acceptable or indicates a good level of consistency in the comparative judgements represented in that matrix. In contrast, if CR is more than 0.1, inconsistency of judgements within that matrix has occurred and the evaluation process should therefore be reviewed, reconsidered, and improved.

2.3

Determination of the evaluation set

For recording students' learning in practical course conveniently, the model adopts 5 class marking system that including excellent, good, medium, pass, fail. The corresponding score: excellent: 95, good: 85, medium: 75, passed: 65, failed: 50. The evaluation set is defined as Y= {Y₁, Y₂, Y₃, Y₄, Y₅} = {95, 85, 75, 65, 50}. The appraisal set is defined as an appraisal vector:

where r_mn is the fuzzy membership degree of appraisal of the factor X_m to

3.

APPLICATION OF THE MODEL

The model has been applied to integrated course project for circuit and electronic. The fuzzy judgment matrix X and X_m were obtained by pair-wise comparison, the fuzzy judgement vectors are as follows:

The relative weights of each element from the first achievement degree evaluation’s factor and the second achievement degree evaluation’s factor and the consistency ratio of each matrix were analyzed, as follows:

W = (0.127, 0.296, 0.228, 0.199, 0.027, 0.047, 0.077)^T, CR=0.047.

W₁ = (0.103, 0.292, 0.605)^T, CR₁=0.05. W₂ = (0.344, 0.535, 0.121), CR₂=0.023.

W₃ = (0.225, 0.696, 0.079)^T, CR₃=0.092. W₄ = (0.162, 0.309, 0.529)^T, CR₄=0.011.

W₅ = (0.137, 0.493, 0.37)^T, CR₅=0.04. W₆ = (0.103, 0.605, 0.292), CR₆=0.037.

W₇ = (0.575, 0.114, 0.311)^T, CR₇=0.096.

Since all CR < 0.1, indicates a good level of consistency in the comparative judgements represented in that matrix and the pair wise comparison matrix can be accepted.

For convenience of statistics and calculation at the teaching process, the weights were rounded, and expressed as percentages.

W = (10%, 30%, 20%, 20%, 5%, 5%, 10%)^T,

W₁ = (10%, 30%, 60%)^T, W₂ = (35%, 50%, 15%)^T, W₃ = (20%, 70%, 10%)^T,

W₄ = (20%, 30%, 50%)^T, W₅ = (10%, 40%, 50%)^T, W₆ = (10%, 60%, 30%)^T,

W₇ = (60%, 10%, 30%)^T

Take a student as an example: the achievement degree of this student in the whole teaching process as shown in Table 1. The appraisal of the second achievement degree evaluation index factors were calculated by relation equations B_i = W_i · R_i, the results as shown in Table 4.

Table 4.

The appraisal of the second achievement degree evaluation of a student

The final fuzzy evaluation vector B calculated by relation equations B = W · R, the results as shown in Table 5.

Table 5.

The final fuzzy evaluation of the student

Further, quantification of result can be obtained by .

4.

CONCLUSION

This paper choose the evaluation index from Engineering Education Accreditation Standards and establishes a process evaluation model of students' learning in practical course by the fuzzy comprehensive evaluation method and fuzzy analytical hierarchy process. Take integrated course project for circuit and electronic and a student as an example, the procedure of model establishment is introduced. It can be concluded that the model could accurate, clear reflection of the performance of student in practical course, facilitate teacher assessing the student more objective and fair. This model could be extended to the graduation thesis, production internships and other practical courses.